Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities

TL;DR

This work establishes that the two-loop chiral superstring measure is fully slice-independent and unambiguous by analyzing gravitino slices supported at two points and demonstrating independence from the slice locations $x_eta$, as well as from auxiliary insertions $q_eta$ and $p_a$. By deriving explicit delta-function gravitino expressions for the correction terms ${ m X}_i$, and carefully handling the ${ m X}_6$ contribution, the authors prove a holomorphic, coincedence-regular measure $oldsymbol{igl[}oldsymbol{ abla}igr]$ that remains well-defined under all limits, including $q_eta o p_a$, $x_eta o q_eta$, and $x_1 o x_2$. The crucial achievement is showing that the potentially singular limit $x_eta o q_eta$ yields a correct and well-defined picture-changing operator $Y(z)= frac{}{}igl[ frac{}{} ext{delta}(eta(z))S(z)igr]$, resolving long-standing ambiguities in the BRST construction. The results pave the way for representing the measure in terms of modular forms in the follow-up work, providing a robust, unambiguous foundation for two-loop superstring amplitudes.

Abstract

The chiral superstring measure constructed in the earlier papers of this series for general gravitino slices is examined in detail for slices supported at two points x_α. In this case, the invariance of the measure under infinitesimal changes of gravitino slices established previously is strengthened to its most powerful form: the measure is shown, point by point on moduli space, to be locally and globally independent from the points x_α, as well as from the superghost insertion points p_a, q_αintroduced earlier as computational devices. In particular, the measure is completely unambiguous. The limit x_α= q_αis then well defined. It is of special interest, since it elucidates some subtle issues in the construction of the picture-changing operator Y(z) central to the BRST formalism. The formula for the chiral superstring measure in this limit is derived explicitly.